Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

learn more… | top users | synonyms

0
votes
2answers
54 views

Is that true that the normalizer of $H=A \times A$ in symmetric group cannot be doubly transitive?

Is that true that the normalizer of $H=A \times A$ in the symmetric group cannot be doubly transitive where $H$ is non-regular, $A \subseteq H$ and $A$ is a regular permutation group?
3
votes
1answer
36 views

Finite abelian groups (application of structure theorem)

Problem Find all finite abelian groups that simultaneously have exactly $7$ elements of order $2$, exactly8 elements of order $3$, exactly $8$ elements of order $4$, at least an element of order ...
5
votes
1answer
47 views

Group of order $p^{n}$ has normal subgroups of order $p^{k}$

Q: Prove that a subgroup of order $p^n$ has a normal subgroup of order $p^{k}$ for all $0\leq k \leq n$. Attempt at a proof: We proceed by Induction. This is obviously true for $n=1, 2$. Suppose it ...
3
votes
3answers
57 views

A group action proof without group actions?

I am currently teaching an undergraduate abstract algebra course out of Saracino, Abstract Algebra: A First Course. Exercise 13.13 asks the following: Let $K$ be the subgroup $\{ e, (1, 2)(3, 4), ...
0
votes
0answers
30 views

Sylow Theorems for Symmetric (Permutation) Groups

The General Linear Group $GL(n,\mathbb{F}_p)$ has an interesting property that the proof of Sylow theorem for this group can be given which is based on the natural action of this group on the ...
1
vote
1answer
33 views

Non-existence of non-abelian simple groups of order $90$ by counting elements [duplicate]

I want to show that no group of order $90=2 \cdot3^2 \cdot5 $ is simple. Part (iii) of Sylow's Theorem gives $$n_2 \in\{1,3,5,9,15,45\} $$ $$n_3 \in \{1,10\} $$ $$n_5 \in \{1,6\} $$ I am aware of ...
1
vote
0answers
25 views

If $\sigma$ is a cycle of length $r$, then it has order $r$?

I should specify that $\sigma \in S_n$, the symmetric group. I've written down some permutations and it seems like their order correspond to their lengths. How can I prove this? I was thinking of ...
-1
votes
0answers
27 views

Module of representation matrix

Can someone please show me why the module of any representation matrix in a one-dimensional representation of a finite group is equal to 1? and please define module of a representation as well. ...
0
votes
3answers
46 views

elements of order $3$ in $A_n$

Let $S_n$ be the permutation group on $n$. We know that $A_n \trianglelefteq S_n$. How many elements $ a \in A_5$ have order three. Is there any formula for finding number of elements in $ S_n $ or ...
0
votes
1answer
26 views

Find number of all $a \in G $ such that $o(a) =3$

Let $G$ be a group and $|G|= 51$ find number of all $a \in G$ such that $o(a)=3$ My solution : by this theorem : if $|G|=pq$ that $ p ,q$ are prime. If $ q\nmid p-1 $ then $\quad$ $G \cong \Bbb ...
0
votes
2answers
39 views

Group theoretical proof of $\varphi(rs)=\varphi(r)\varphi(s)$ through generators of the group.

Given a group $G=\langle a\rangle$ of order $rs$, with $(r,s)=1$, I showed there exist unique $b,c\in G$ such that $a=bc$ with $b$ of order $r$ and $c$ of order $s$. The latter is a direct consecuense ...
2
votes
0answers
27 views

Characterize all $\mathbb Z[i]$ modules of order $21$ and $65$

I am trying to figure out how many $\mathbb Z[i]$ modules (up to isomorphisms) are of $21$ and $65$ elements. I've done a few similar exercises for finite abelian groups ($\mathbb Z$ modules) but I am ...
1
vote
4answers
122 views

How many ways can the vertices of an equilateral triangle be colored using three different colors?

Its not $3^3$ because some of the colorings are equivalent. How would I apply Burnside's theorem to this?
15
votes
1answer
244 views

endomorphism of finite groups

Have $\mathcal{G}$ denote the set of finite groups with at least $2$ elements. How would I go about showing that if $G \in \mathcal{G}$, then $\left|\text{End}(G)\right| \le ...
1
vote
1answer
34 views

Primitive Permutation Group and Centralizers

Automorphism group of the Alternating Group - a proof In the above question, Derek Holt asserts that a primitive permutation groups has trivial centraliser in the symmetric group. Since I could not ...
0
votes
0answers
33 views

Does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$?

For $p$ prime, does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$? (It does seem to be the case for $p=3,5,7,11$) If yes, what are the generators? I guess this is simple, but my ...
2
votes
0answers
51 views

Every non-regular minimal normal subgroup of a doubly transitive group is primitive and simple

Prove that every non-regular minimal normal subgroup $N$, of a doubly transitive permutation group $G$, is primitive and simple. I proved that $N$ is primitive; but how I can prove that $N$ is ...
0
votes
0answers
33 views

Order of center of a p-group deduce abelian [duplicate]

Let $G$ be a group of order $p^n$, $p$ a prime. Suppose the center of G has order at least $p^{(n−1)}$. Prove that G is abelian. Attempt: use the class equation $|G|=|Z(G)|+ \sum_{i \in ...
1
vote
1answer
46 views

Embedding $S_n$ in $A_{2n}$

I want to embed the symmetric group $S_n$ into the bigger alternating group $A_{2n}$. How could I find such an injective homomorphism?
3
votes
1answer
42 views

Galois groups of quintics

I'm trying to determine which subgroups of $S_5$ occur as the Galois group of an irreducible quintic $f\in\Bbb{Z}[X]$. I know such a subgroup of $S_5$ should be transitive, leaving only five ...
1
vote
2answers
57 views

Number of non-isomorphic groups of order $p^2$

The number of non-isomorphic groups of order $p^2$, where $p$ is a prime number is: 1. 1 2. $p$ 3. 2 4. $p^2$ What is simplest method to find number of non-isomorphic group? I read from various ...
1
vote
2answers
37 views

Composition series of nilpotent group

I found this problem in The Theory of Groups by Marshall Hall. Let the group $G$ be of order $p^rq^s$. If $G$ has two composition series $1 \unlhd A_1 \unlhd A_2 \unlhd \cdots \unlhd A_r \unlhd ...
2
votes
0answers
20 views

Direct product of groups and isomorphism [duplicate]

Let $A, B, C$ three groups such that $A \times C \cong B \times C$. I already know that if $A, B$ and $C$ are abelian and finite, then $A \cong B$. I think this result does not hold anymore if they ...
3
votes
2answers
31 views

Number of $n-1$-dimensional subspaces of $n$-dimensional space over finite field

I got a question with two parts. Let $V$ be a $n$-dimensional vector space over $\mathbb{F}_{p}$ - finite field with $p$ elements. a) How many $1$-dimensional subspaces $V$ has. b) How many ...
0
votes
1answer
35 views

Isomorphism of Quaternion group

Prove that $Q_{8} \cong H \rtimes G \Rightarrow H=\{e\}$ or $G=\{e\}$. My proof (is it correct?): We know that (it's a fact from the lecture): If $M \cong H\rtimes G$, then $H \cong K \unlhd M$ ...
1
vote
2answers
39 views

Isomorphism type of a finite group with respect to multiplication modulo 65

I'm the same guy revising for my group theory exam and posted a few days ago. I'm at the chapter on Finitely Generated Abelian Groups, and my prof gave this example which I don't quite understand: ...
6
votes
1answer
49 views

Exercise on Induced Representations of 1 dimensional complex representation

I'm having a hard time trying to solve the following problem coming from Eingof's book "Introduction to representation theory" (page 55 of the book in PDF format ...
0
votes
1answer
25 views

if $G$ is a finite group and every sylow subgroup of it is cyclic,prove that $G$ is super-solvable.

if $G$ is a finite group and every sylow subgroup of it is cyclic,prove that $G$ is super-solvable. we have that $G$ is solvable,I want to show that all factors of derived series are cyclic. but no ...
0
votes
0answers
45 views

group without involution is 2-divisible

Let $G$ be an arbitrary torsion group without involutions. Show that $G$ is 2-divisible. I think it is enough to show $G$=$2G$ but i can't show why $2G$ can't be proper subgroups of $G$ ? Please ...
3
votes
2answers
59 views

Existence of a normal subgroup in G

Today on my algebra test I had such an exercise: Let $|G|=66$. Show that there is a normal subgroup in $G$ of order $3$. I am not even sure that's true. I wanted to show that $n_{3}$=1. But from ...
1
vote
0answers
20 views

Commutators inside center of factor subgroups

Let $G$ be a finite group. Assume that $K \subseteq L \trianglelefteq G$ with $K \trianglelefteq G$. Then $L /K \subseteq \textbf{Z}(G / K)$ if, and only if $[G,L] \subseteq K$. I know it is related ...
1
vote
1answer
70 views

Automorphism group of the Alternating Group - a proof

I was trying to read the following lemma which admit as an easy corollary the structure of the automorphism group of the alternating group on $n\geq 7$ elements. Anyway there are two points that ...
0
votes
1answer
24 views

Existence of integer $n > 2$ such that for any abelian group $G$ , $G_n:=\{e\} \cup \{a \in G :o(a)=n \} $ is a subgroup of $G$ [closed]

Does there exist an integer $n > 2$ such that for any abelian group ( or at-least any finite abelian group ) $G$ , the set $G_n:=\{e\} \cup \{a \in G :o(a)=n \} $ is a subgroup of $G$ ?
1
vote
0answers
20 views

Prove that $Q_{8}'=\{1,-1\}$. Is my proof correct?

Prove that Prove that $Q_{8}'=\{1,-1\}$. My proof: $Q_{8}'\neq \{1\}$, because $Q_{8}$ is not abelian. $|Q_{8}/\{-1,1\}|=4$. So $Q_{8}/\{-1,1\} \cong \mathbb{Z}_4$ or $Q_{8}/\{-1,1\} \cong ...
1
vote
1answer
26 views

Showing that G is solvable

Let $|G|=200$. Show that G is solvable. My beginning of the proof: $|G|=200=2^3*5^2$ Let $n_5$ be the number of Sylow p-subgroups. Then $n_5|8$ and $n_5\equiv1 mod5$. And it implies that $n_5=1$. ...
1
vote
2answers
24 views

Injective Homomorphism from a group into $GL_n$

$|G|=n\ge 2<\infty,$ A group, I need to know which of the followings are true? $\exists$ allways an injective homomorphism from $G$ into $S_n$ $\exists$ allways an injective homomorphism from $G$ ...
0
votes
1answer
43 views

Trace group of a skew group algebra of a commutative domain

Let $R$ be a commutative noetherian domain that is also an algebra over a field $k$ Let $G$ is a finite group that acts on $R$ in a non-trivial way. Let $A=R*G$ be the skew group algebra of this ...
1
vote
0answers
28 views

Generating a group by its $q$-elements.

Let $G=PQ$ be a solvable group where $P$ and $Q$ are $p$-subgroup and $q$-subgroup of $G$ respectively. Also suppose that $Q$ is not normal in $G$. Is it true that the group generated by all ...
2
votes
0answers
53 views

If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ?

If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ? ( By Cauchy's theorem I can show that there are elements of order $2$ and $3$ but cant proceed ...
0
votes
0answers
30 views

Intersection theorems for a certain type of subsets of integers modulo $N$

I've been working on something with integers modulo $N$ and have sort of hit a roadblock where I'd like to have some references. The particular problem goes as follows. We have a system $\mathcal{S}$ ...
1
vote
1answer
32 views

Existing of such automorphism subgroup

Let $G$ be abelian group and $Aut(G)$ be the automorphism group of $G$ , I am looking for a nontrivial subgroup $H$ of $Aut(G)$ such that $gcd(|H|,|G|)=1$. Does such subgroups always exit ?
2
votes
1answer
15 views

semi-direct product of subgroups of $D_6$

I have a hexagon with edges $A,B,C,D,E,F$ and its symmetry group $D_6$. I want to prove that $D_6 = H \rtimes M$ given the subgroups $H = \{ g \in D_6 \,\,|\,\, g \text{ permutes } \{A,C,E\} \}$ and ...
1
vote
1answer
21 views

$D_6$ and cycle notation problem

I have a hexagon with edges $A,B,C,D,E,F$ and I want to work with its symmetry group $D_6$ in cycle notation. My calculations don't yield consistent results. For example, I correctly get $r^4 \cdot ...
0
votes
0answers
23 views

Can a non-abelian group of order $105$ have trivial center ?

Can a non-abelian group of order $105$ have trivial center ?
1
vote
1answer
92 views

Group and Ring Homomorphisms between $\mathbb{Z}_p$ and $\mathbb{Z}_{2p}$

Let $p$ be prime with $p > 2$. (a) Determine the number of group homomorphisms between $\mathbb{Z}_p$ and $\mathbb{Z}_{2p}$ (b) Determine the number of ring homomorphisms between $\mathbb{Z}_p$ ...
2
votes
1answer
45 views

Why is $D_5$ a subgroup of the icosahedral group

According to Wikipedia $D_5$ is a subgroup of the group of rotational symmetries of an icosahedron: http://en.wikipedia.org/wiki/Icosahedral_symmetry. I know this isn't very rigorous, but intuitively ...
0
votes
1answer
36 views

Symmetric polynomials, group of permutations

Could somebody give me a clue, related to the possible solution of the problem? Let's denote a polynomial $f(x_{1}, x_{2}, x_{3}, x_{4}, x_{5})=x_{1} x_{2} x_{3} +x_{2} x_{3} x_{4} + x_{3} x_{4} ...
0
votes
1answer
21 views

Subgroup lattice of $U(12)$

$U(12)$ is not cyclic. Order of $U(12)$ is $4$. By Lagrange's Theorem, order of a subgroup must divide the order of the group. Hence any subgroup of $U(12)$ must have order $1, 2 \text{ or } 4$. ...
0
votes
2answers
38 views

Order of $A \in GL(n,\mathbb Z_p)$ cannot exceed $p^n-1$ ?

If $A \in GL(n,\mathbb Z_p)$ then is it true that order of $A$ cannot exceed $p^n-1$ ?
0
votes
0answers
39 views

Determine a set of coset representatives

Let $M$ be a $2\times2$ invertible matrix with entries in $\mathbb{Q}$. Let $$\mathbb{Z}^2(M):= \mathbb{Z}^2\cap (M^{-1}\mathbb{Z}^2M).$$ It is clear that $\mathbb{Z}^2(M)$ is a subgroup of ...